Abstract:
The single-index model (SIM) is an efficient and widely used regression method, especially
when dealing with high-dimensional data. As a typical case of the projection pursuit
regression, the single-index model can avert a specific problem of “the curse of dimension-
ality” in nonparametric models and depict the relationship between the response variable
and the covariates straightforwardly.
In this thesis, we apply a novel nonparametric prior in the estimation of the link function
in the SIM. Instead of a mixture of beta densities, we use a mixture of B-spline densities
with their weights and knots generated by two Dirichlet processes. Based on the Bayesian
median regression, the asymmetric Laplace distribution (ALD) is chosen as the error dis-
tribution, and the latent binary indicator variables are applied to help select the significant
predictors. The posterior samples are generated by a Metropolis-Hastings-within-Gibbs
algorithm.
To investigate the performance of the B-spline prior, we use numerical studies to show its
flexibility and adequate coverage of the estimates compared to the Bernstein polynomial
prior when facing complex single-index models. Finally, we put the B-spline prior into
an application of the sewage treatment data, which indicates the outperformance of the
B-spline prior in real data analysis.
